# Cosine of Complement equals Sine

## Theorem

$\map \cos {\dfrac \pi 2 - \theta} = \sin \theta$

where $\cos$ and $\sin$ are cosine and sine respectively.

That is, the sine of an angle is the cosine of its complement.

## Proof 1

 $\displaystyle \map \cos {\frac \pi 2 - \theta}$ $=$ $\displaystyle \cos \frac \pi 2 \cos \theta + \sin \frac \pi 2 \sin \theta$ Cosine of Difference $\displaystyle$ $=$ $\displaystyle 0 \times \cos \theta + 1 \times \sin \theta$ Cosine of Right Angle and Sine of Right Angle $\displaystyle$ $=$ $\displaystyle \sin \theta$

$\blacksquare$

## Proof 2

 $\displaystyle \cos \left({\frac \pi 2 - \theta}\right)$ $=$ $\displaystyle \cos \left({\theta - \frac \pi 2}\right)$ Cosine Function is Even $\displaystyle$ $=$ $\displaystyle \sin \left({\theta - \frac \pi 2 + \frac \pi 2}\right)$ Sine of Angle plus Right Angle $\displaystyle$ $=$ $\displaystyle \sin \theta$

$\blacksquare$

## Proof 3

 $\displaystyle \map \cos {\dfrac \pi 2 - \theta}$ $=$ $\displaystyle \frac 1 2 \paren {e^{i \paren {\frac \pi 2 - \theta} } + e^{-i \paren {\frac \pi 2 - \theta} } }$ Cosine Exponential Formulation $\displaystyle$ $=$ $\displaystyle \frac 1 2 \paren {e^{i \frac \pi 2} e^{-i \theta} + e^{-i \frac \pi 2} e^{i \theta} }$ Exponential of Sum: Complex Numbers $\displaystyle$ $=$ $\displaystyle \frac 1 2 \paren {\paren {\map \cos {\frac \pi 2} + i \, \map \sin {\frac \pi 2} } e^{-i \theta} + \paren {\map \cos {-\frac \pi 2} + i \, \map \sin {-\frac \pi 2} } e^{i \theta} }$ Euler's Formula $\displaystyle$ $=$ $\displaystyle \frac 1 2 \paren {i e^{-i \theta} - i e^{i \theta} }$ Cosine of Right Angle, Sine of Right Angle, Cosine Function is Even, Sine Function is Odd $\displaystyle$ $=$ $\displaystyle \frac 1 {2 i} \paren {e^{i \theta} - e^{-i \theta} }$ Definition of Imaginary Unit $\displaystyle$ $=$ $\displaystyle \sin \theta$ Sine Exponential Formulation

$\blacksquare$

## Historical Note

The result Cosine of Complement equals Sine was discovered and documented by Varahamihira in the $6$th century CE.